COUNTING BPS STATES IN CONFORMAL GAUGE THEORIES Alberto Zaffaroni PISA, MiniWorkshop 2007 Butti, Forcella, Zaffaroni hepth/0611229 Forcella, Hanany, Zaffaroni hepth/0701236 Butti,Forcella,Hanany,Vegh, Zaffaroni, to appear
Counting problems in N=1 supersymmetric gauge theory are an old and vast subject: There are various type of partition functions for BPS states: ½ BPS states = Chiral Ring ¼ BPS states Supersymmetric Index Study of the moduli space; generators for the chiral ring and their relations Dependence of the partition function on the coupling Statistical properties of the BPS states and relation to black holes entropies
The Chiral Ring Interest in chiral primary gauge invariant operators: Q O= 0 α O O +Q (...) α A product on chiral primaries is defined via OPE The set of chiral primaries form a ring Expectation values and correlation functions do not depend on position.
In supersymmetric gauge theories with chiral matter superfields X, vector multiplets W and superpotential chiral gauge invariant operators are combinations of however constrained by W n n n Tr (X ), Tr (X W α), Tr (X WαW β) 1. Finite N size effects 2. F term relations Tr X N N+1 i itr i=1 (X )= c (X ) W = DD (X) 0 Classical relations may get quantum corrections Appearance of W included in a superfield structure
GENERAL PROBLEM: count the number of BPS operators According to their global charge: n ( N ) = number of BPS operators k g ( ) ( ) k N a = nk N a with charge k a is a chemical potential for global and R charges N is the number of colors Note: the number of gauge invariant operators is infinite However these typically have charges under the global symmetries and the number of operators with given charge is finite
For N=4 SYM the problem is very simple: 3 adjoint fields Φ F-terms [ Φ, Φ ] = 0 i i j 3 commuting adjoint matrices can be simultaneously diagonalized: g(n) is the generating function for symmetric polynomials in the eigenvalues For a generic N=1 gauge theory the problem is hard: non trivial F term relations for many fields finite N relations
How to compute gauge invariants and generating functions: The problem of finding gauge invariants goes back to the ninenteenth century. In mathematics this is invariant theory. N N matrices X R[X ] = [X ]/{ W(X ) = 0} ij ij ij ij R INV = R[X ] // G ij General methods due to Hilbert: free resolutions, syzygies Now algorithmical (Groebner basis) With computers and computer algebra programs really computable (but for small values of N) Still very hard to get general formulae for generic N
The problem drastically simplifies for the class of superconformal gauge theories with AdS dual: AdS5 H D3 branes probing Calabi-Yau conical singularities Four dimensional CFTs on the worldvolume Connection provided by the AdS/CFT correspondence
D3 branes probing a conical Calabi-Yau with base H: The near horizon geometry is AdS5 H The worldvolume theory is a 4d conformal gauge theory CY condition implies that H is Sasaki-Einstein. H = S, T, Y, L 5 1,1 pq, pqr,, Few metrics known ( ) Many interesting question solved without knowledge of the metric
EXAMPLES: 3 5 C=C(S) N=4 SYM A 1,1 C(T ) Conifold B W = εε AB A B ij pq i p j q
C 3 / Z 3 U V Orbifold Projection Of N=4 SYM W = ε UVW W ijk i j k 152 CL ( ) W =...
General properties: SU(N) gauge groups adjoints or bi-fundamental fields superpotential terms = closed loops in the quiver An infinite class of superconformal theories that generalize abelian orbifolds of N=4 SYM
General properties: The moduli space of the U(1) theory is the CY The moduli space of the U(N) theory is the symmetrized product of N copies of the CY: N branes: The fact that the gauge group is SU(N) implies the existence of baryons in the spectrum
General properties: Classification in terms of global charges: Non anomalous abelian symmetries in the CFT: r flavors (R) symmetries s baryonic symmetries CY isometries RR fields r is at least 1 r=3 r = 1,2,3 R symmetry 3 isometries=toric CY
DIGRESSION: Correspondence between CY and CFT Comparison with predictions of the AdS/CFT Connections to dimers: (Okunkov,Nekrasov,Vafa Hanany,Kennaway) Geometric computation without metric (Martelli,Sparks,Yau) AdS/CFT, combinatorics, a-maximization
Connection to dimers: Okounkov,Nekrasov,Vafa Franco,Kennaway,Hanany,Vegh,Wecht 152 L
Dimers, combinatorics and charges: Hanany-Witten construction for local CY 21 Y delpezzo 1 = Y 21
Connection to a-maximization: Central charge of the CFT determined by combinatorial data: 9 Tr 3 V,V,V i j k i, j, k a= R = < > aa a 32 i j k d i = 1 a i = 2 Butti,Zaffaroni Benvenuti,Pando-Zayas,Tachikawa Lee,Rey Thanks to a-maximization (Intriligator,Wecht) the exact R-charge of the CFT is obtained by maximizing a
The BPS spectrum of states: the N=1 chiral ring Consider the cases: N=4 Conifold Mesonic operators: k Tr( Φ ) n Tr(AB) Baryonic operators: Det(A) Det(B) All gauge invariant single and multi trace operators subject to F term conditions: N=4 [ Φ, Φ ] = 0 i j Conifold A i B p A=A j j B p Ai
Warming up: N4 SYM n m p Tr( ΦΦΦ ) Focus on single trace operators: 1 2 3 [ Φ, Φ ] = 0 i j Φ1 q1 Φ2 q2 Φ3 q3 Generating function: 1 1 gq () = (1 q)(1 q)(1 q) 1 2 3
Warming up: the conifold Focus on single trace operators: Tr(Ai B 1 j...a 1 i B k j ) k A i t1 A i B p A=A j j B p Ai B i t 2 Generating function: 2 n n 1( 1, 2) = ( + 1) 1 2 n= 1 g t t n t t
Not the smart way of computing: Moduli space D3 branes CY Two equivalent moduli space parameterizations: VEV of elementary fields (modulo complexified gauge transformations) Chiral gauge invariant operators: set of holomorphic functions on the moduli space (CY).
Example: N4 SYM Holomorphic functions on : 1 2 3 C 3 n m p n m p z z z ΦΦΦ 1 2 3 Tr( ) Generating function: 1 gq 1() = (1 q)(1 q)(1 q) 1 2 3 q 2 q 1 q 3 Mesonic operators have zero baryonic charge: only 3 independent charges q
Toric Calabi-Yau: Obtained by gluing copies of 3 C g ( q) 1 = I 1 I I I (1 q )(1 q )(1 q ) n m p 1 2 3 CONIFOLD [Martelli,Sparks,Yau]
Full set of operators and their dual interpretation Mesonic operators: Single traces = holomorphic function on the CY index theorem(martelli,sparks,yau) Multi traces = for N colors, holomorphic functions on N the symmetric product Sym(CY) (Benvenuti,Feng,Hanany,He) Bulk KK states in ADS: gravitons and multigravitons Baryonic operators: Determinants = operators with large dimension (N) Solitonic states in AdS: D3 branes wrapped on 3 cycles
Subtelties also for mesons: Mesonic operators in the bulk are KK states Dimension of order N Giant gravitons Tr N N+1 i itr i=1 (X )= c (X ) D3 wrapping trivial 3 cycles stabilized by rotation
QUANTIZING SUPERSYMMETRIC D3 BRANE STATES 1. It contains at once all BPS states 2. It allows a strong coupling computation (AdS/CFT) Consider a D3 brane wrapped on a 3 cycle in H, including excited states, possibly non static Branes on trivial cycles stabilized by flux+rotation =Giant Gravitons mesons Branes on non-trivial cycles, possibly excited and non static baryons
SUPERSYMMETRIC CLASSICAL D3 BRANES CONFIGURATIONS Σ D3 brane on cycle in H: Divisor C( ) in the CY Σ (Σ,t) it wraps it wraps (rotating t to r in AdS) (Σ,r) Σ H SUPERSYMMETRIC D3 BRANE = holomorphic surface in CY
Translation to geometrical problem: count all holomorphic surfaces in a given equivalence class of divisors: An holomorphic surface is locally written as an equation in suitable complex variables Mesons: trivial divisor zero locus of holomorphic functions n m p P(z i)= anmpz1z2z3 Baryons: non trivial divisors sections of suitable line bundles baryonic charge B=degree of the line bundle P(z i) 0 H (X,O(B))
Computing the generating function for BPS D3 brane states: Classical BPS D3 brane states identified with holomorphic surfaces of class B (line bundle sections), computed using index theorem g ( q) = 1, B n m p I q1 q2 q3 q q (1 )(1 )(1 ) I I I I CONIFOLD
QUANTIZATION OF THE CLASSICAL CONFIGURATION SPACE OF SUPERSYMMETRIC D3 BRANES Done with geometric quantization [Beasley]: Full Hilbert space at fixed baryonic charge B obtained from N=1 result by taking N-fold symmetrized products of sections Pi 0 H (X,O(B)) P 1,P 2,...,P N > υ g ( q) = Exp( g ( q ) υ / k) N k k BN, k = 1 1, B (counts symmetrized products) Also known as pletystic exponential
Count symmetrized products of elements P in a set S with generating function ( ) n g1 q = q Introduce a new parameter: υ n S 1 N g( q, υ) = = υ gn ( q) (1 ) n S n υ q N = 1 n k kn k k υ = υ = 1 υ n S k= 1 n S k= 1 Exp( log(1 q )) Exp( q / k) Exp( g ( q ) / k)
Example: the conifold Geometry: the conifold can be written as a quotient: four complex variables modded by a complex rescaling λ *, ( x xλ, x x / λ, x x λ, x x / λ) 1 1 2 2 3 3 4 4 charges (1,-1,1,-1) Similar to projective space. Homogeneous coordinates exist for all toric manifold Holomorphic surfaces can be written as P ( x 1,..., x 4 ) = 0
Example: the conifold Field Theory: four charges, one R, two flavors, one baryonic x x 1 3 x 2 x 4 Homogeneous rescaling = baryon number
Mesonic operators (charge B=0) g ( ) 1, B 0 q P( x ) = x x + x x +... i counts operators 1 2 3 4 Tr (A B...A B ) i j i j = 1 1 m m (m+1,m+1) with charge t1 for A and t2 for B g ( t, t ) = ( m + 1) t t 2 m m 1, B = 0 1 2 1 2 m = 0 g ( ), = 0 q NB counts multitraces k m Tr(AB)...Tr(AB)
Baryonic operators at charge B=1 P( x ) = x + x + x x +... i 2 1 3 1 2 g1, B ( q ) = counts operators (m+2,m+1) g ( t, t ) = ( m+ 1)( m+ 2) t m t m with charge t1 for A and t2 for B 1, B= 1 1 2 1 2 m= 0 gnb, ( q) counts determinants
Comments: Not all baryonic operators are factorizables as: Det (A) Mesons For example there are 2(N-1) non factorizable components of: Det(ABA,A,...,A)
Full partition function for the conifold N=1 Finite N: Checked against explicit computation for N=2,3.
Structure of the partition function: N=2: 10 generators (with relations): Det(A)=εεA A Det(B) Tr(AB) i j 3 3 4 General N: (N+1,1) generators Det(A) (1,N+1) generators Det(B) (n,n) generators Tr(AB) n=1,,n-1 n 2(N-1) generators Det(ABA,A,...,A)... other non factorizable baryons
Comments I Similar analysis can be done for other CY: A general lesson on the structure of BPS partition functions: N=1 result decomposes in sectors with fixed baryonic charge B The finite N result for baryonic charge B is obtained by PE (symmetrized products) from N=1 result The full partition function for finite N is obtained by resumming the contribution of fixed baryonic charge
Comments II ½ BPS partition functions seem independent of the coupling constant: strongly coupled AdS/CFT computation agrees with weakly coupling analysis Relation of baryonic charge sectors with discretized Kahler moduli of the geometry Similarity of results with topological strings/nekrasov partition functions
CONCLUSION Intriguing interplay between geometry and QFT AdS/CFT: index theorem and localization QFT computation: invariant theory Other interesting questions: Partition functions for general CY Index and ¼ BPS states Non CY vacua Termodinamic properties of partition functions
Toric case is simpler: Geometrically: toric cones are torus fibrations over 3d cones. 3 U (1) 2 U (1) U (1) APPENDIX: TORIC CY 43
All information on toric CY: convex polygon with integer vertices. Conifold toric diagram APPENDIX: TORIC CY 44
21 Y APPENDIX: TORIC CY 45
Global charges and geometry: There are d abelian symmetries in the CFT: 1 R symmetry 2 flavor symmetries d-3 baryonic symmetries isometries reduction of RR potentials on the d-3 3-cycles d = number of vertices d-3 three cycles d 1 2 APPENDIX: TORIC CY 46 3
Toric case is simpler: Gauge theory: constraints on number of fields: G + V = F G = number gauge groups F = number of fields V = number of superpotential terms Conformal invariance requires linear conditions on R-charges G beta functions conditions V superpotential conditions for F=V+F fields IR fixed point APPENDIX: TORIC CY 47
Comment: Conformal invariance conditions have d-1 independent solutions d-1=number of global non anomalous abelian charges R charges of the F elementary fields can be d expressed in terms of d charges with a ai i = 1 i=1,,d i = 2 Global charges are associated with vertices APPENDIX: TORIC CY 48
Example: conifold d=4 vertices d-3=1 three cycles Four charges: one R, two flavors, three isometries one baryonic RR 4-form on 3-cycle APPENDIX: TORIC CY 49
Example: conifold 1 2 1 A 1 2 2 1 2 B 1 2 1 [Franco-Hanany-Kennaway-Vegh-Wecht] [Feng-He-Kennaway-Vafa] APPENDIX: TORIC CY 50
Example: conifold 1 2 1 A 1 2 2 11 2 B 1 2 1 APPENDIX: TORIC CY 51
Example: conifold A 1 2 1 1 2 2 1 2 B 1 2 1 APPENDIX: TORIC CY 52
Example: conifold A 1 2 B APPENDIX: TORIC CY 53
Example: conifold a 3 zig zag path Global charges are associated with vertices APPENDIX: TORIC CY 54
The full multitrace contribution for N colors is given in terms of the result for N=1: N branes N Holomorphic Functions on (Sym(CY) ) υ g ( q) = Exp( g ( q ) υ / k) N k k N k = 1 1 (counts symmetrized products) Known also as the pletystic exponential g ( q) 1 is the generating function for holomorphic functions [Benvenuti,Feng,Hanany,He] APPENDIX: TORIC CY 55